Obstacle problem for nonlinear integro-differential equations arising in ...

Ricerche di Matematica 56: 1–17 (2007) DOI: 10.1007/s11587-007-0001-x

Anna Lisa Amadori

Obstacle problem for nonlinear integrodifferential equations arising in option pricing

Received: February 21, 2006 / Accepted: July 20, 2006

Abstract We study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate, which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle. We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity. Keywords Integro-differential equations · Obstacle problem · Viscosity solutions · American options Mathematics Subject Classification (2000) 45K05 · 35K85 · 49L25 · 91B24 1 Introduction This paper is devoted to the study of obstacle problems of type min ∂t u + F(x,t, u, I u, Du, D2u), u − uo = 0, (x,t) ∈ R N ×(0, T ], u(x, 0) = uo (x, 0),

x ∈ RN ,

(1) (2)

where F is degenerate elliptic, quasi-monotone with respect to u, and monotone with respect to I u. Here, I u stands for an integral term given by M u(x + z,t), u(x,t) d μx,t (z), (3) I u(x,t) = RN

Communicated by V. Ferone A.L. Amadori (B) Università di Napoli “Parthenope”, Dipartimento di Scienze Applicate, via A. De Gasperi, 5, 80133 Napoli, Italy E-mail: [email protected]

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A.L. Amadori

where μx,t are finite measures, and M is a Lipschitz continuous function, nondecreasing with respect to its first argument. Precise assumptions are postponed to Section 2. This equation models the price of American derivatives in jump-diffusion markets with large investor. In this framework interesting examples concern the semilinear case F(x,t, u, I u, Du, D2u) = LI u + H(x,t, u, I u, Du),

(4)

where H is a nonlinear first order integro-differential operator, and LI is a linear integro-differential operator, possibly strongly degenerate: 1 LI u = − tr σ σ T D2 u + b · Du + c u − d I u. 2 Besides the obstacle grows up exponentially at infinity, namely e−no x uo ∈ C (R N × [0, T ]) ∩ L∞ R N × [0, T ] , for some suitable integer no . The obstacle problem related to integro-partial differential equations have been extensively studied, in the non-degenerate case with linear dependence w.r.t. I u; we mention about this topic [10, 14, 16], and the references therein. With respect to strongly degenerate equations, it is natural to turn to the the theory of viscosity solutions, as well as in [1, 3, 4], all dealing with the Cauchy problem. We also mention [15, 16] concerning obstacle problem related to linear equations, and [2], which studies another type of differential term. In the present paper the results of [3] are extended to the obstacle problem. We begin by establishing that viscosity solutions of the general problem (1), (2) are stable with respect to penalization. Next, in view of dealing with fastly increasing data, we turn to the semilinear case and obtain a strong comparison result. This gives the existence and uniqueness of a solution, which is known to be merely continuous, a-priori. Next uniform estimates of the penalized problems yield the Lipschitz regularity of solutions. Lastly the obtained results are applied to the pricing of American options.

2 Viscosity solutions: penalization method In this section, we show that the penalization of the obstacle actually is a viscosity approximation for (1), (2). Next we shall obtain existence and uniqueness of solutions, by establishing a strong comparison principle. To begin with, we clarify the assumptions that shall be in force in what follows. We consider the same type of integral terms already studied in [1, 3]. Indeed we assume that μx,t are finite positive measures fulfilling

lim

(y,s)→(x,t) R N

ϕ (z)d μy,s (z) =

RN

ϕ (z)d μx,t (z),

(5)

Obstacle problem for nonlinear integro-differential equations arising in option pricing

3

for any continuous function with bounded support ϕ , and that |M(u, v) − M(u , v )| ≤ M (|u − u | + |v − v|) , M(u, u) = 0,

(6)

M(u, v) ≤ M(u , v) if u ≤ u .

We suppose that the obstacle is a continuous function, namely F.0 uo ∈ C (R N × [0, T ]).

Concerning the integro-partial differential operator F, we make the standard assumption of viscosity solutions approach: F.1 F ∈ C (R N × [0, T ] × R × R × R N× S N ) is degenerate elliptic, i.e.

F(x,t, u, I, p, X) ≥ F(x,t, u, I, p, Y))

if X ≤ Y,

and quasi-monotone as a function of u, namely there exists g ∈ C ([0, T ]) with g(0) = 0 and 01 dr/g(r) = +∞, such that F(x,t, u, I, p, X) ≥ F(x,t, v, I, p, X)) − g(u − v)

if u ≥ v.

Here S N stands for the set of symmetrical N × N matrices. In order to handle nonlinearities with respect to the non local term I u, some monotonicity property is needed: F.2 F is non-increasing as a function of I u, namely

F(x,t, u, I, p, X) ≥ F(x,t, u, J, p, X)

if I ≤ J.

We recall for the sake of completeness the definition of viscosity solution for integro-differential problems with non local term I u of type (3), (5), (6). We refer to [8] for a general presentation of the theory of viscosity solutions in the pure differential case, and to [1, 3] for the integro-differential case. Definition 1 Given a function u and (x,t) ∈ R N × [0, T ], we shall say that ∂t u + F(x,t, u, I u, Du, D2 u) ≤ 0 (respectively, ≥ 0) in viscosity sense at (x,t) if one of the following equivalent items holds: 1) τ + F(x,t, u(x,t), I u(x,t), p, X) ≤ 0 (resp., ≥ 0) for all (τ , p, X) in P 2+ u(x,t) (respectively,P 2− u(x,t)); 2) ∂t ϕ (x,t) + F x,t, u(x,t), I u(x,t), Dϕ (x,t), D2ϕ (x,t) ≤ 0 (resp., ≥ 0) in classical sense, for each smooth function ϕ on R N × [0, T ] such that u − ϕ has a local maximum (resp., a local minimum) at (x,t); 3) ∂t ϕ + F x,t, ϕ , I ϕ , Dϕ ,D2ϕ ≤ 0 (resp., ≥ 0) in classical sense, for all smooth functions ϕ touching from above (respectively, from below) the graph of u at (x,t). We recall that the parabolic semi-jets P 2± are defined as

P 2± u(x,t) = (τ , p, X) ∈ R × R N × S N : u(y, s) ≤ ( resp. ≥) 1 u(x,t) + τ (t − s) + p · (x − y) + X(x − y) · (x − y) 2 +o |t − s| + |x − y|2 as (y, s) → (x,t) , Next we define viscosity solutions.

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A.L. Amadori

Definition 2 An upper semicontinuous, locally bounded function u on R N × [0, T ] is a subsolution to (1), (2) if ∂t u + F(x,t, u, I u, Du, D2 u) ≤ 0 in viscosity sense at all (x,t) ∈ R N × (0, T ] such that u(x,t) > uo (x,t), and u(x, 0) ≤ uo (x, 0) for all x ∈ RN . A lower semicontinuous, locally bounded function u is a supersolution to (1), (2) if ∂t u + F(x,t, u, I u, Du, D2 u) ≥ 0 in viscosity sense at all (x,t) ∈ R N × (0, T ], P 2− u(x,t) = 0/ if u(x,t) < uo (x,t), and u(x, 0) ≥ uo (x, 0) for all x ∈ R N . A solution for the problem (1), (2) is any continuous function that is both a subsolution and a supersolution. We emphasize that the two definitions of viscosity sub/supersolution for the obstacle problem are not symmetric. Actually, nothing is imposed to subsolutions on the set u ≤ uo . On the other hand, supersolutions must satisfy u ≥ uo in a dense subset of R N × [0, T ], because the set (x,t) : P 2− u(x,t) = 0/ is dense whenever u is lower semicontinuous. We refer to the book [7] for a detailed proof of this basic result. In particular, it yields that any viscosity solution u satisfies u ≥ uo on R N × [0, T ]. It is well known that semicontinuity preserving properties are crucial for the stability of the notion of viscosity solutions. On the other hand, it has already been noticed in [1] that the integral operator I does not preserve semicontinuity, in general. In the same paper, such difficulty has been overcame by restricting the study to that upper (resp., lower) semicontinuous functions which satisfy the additional property there exists Φ ∈ Bx,t such that for all y, s near at x,t and z ∈ R N M(u(y + z, s), u(y, s)) ≤ Φ (z)

(resp., ≥ −Φ (z)).

(7)

Here the set Bx,t is defined by

Bx,t = Φ ∈C (R N ) ∩ L1 (R N ; μx,t ) :

lim

(y,s)→(x,t) RN

Φ ≥ 0,

Φ (z) d μy,s (z) =

Φ (z) d μx,t (z) .

RN

Indeed, [1, Lemma 1] establishes that I u is upper (resp., lower) semicontinuous whenever u is upper (resp., lower) semicontinuous and satisfies (7). On the other hand, condition (7) does not involve a serious restriction, because it is implied by a suitable rate of growth at infinity: see [3, Lemma 3.1] for a precise statement. In particular, it is trivial if μx,t has compact support for any given (x,t). It is useful to observe that the initial condition can be relaxed, by following [6, Proposition 5]. Definition 3 An upper semicontinuous, locally bounded function u is a subsolution for (1), (2) with generalized initial condition, GIC, if the initial condition u(x, 0) ≤ uo (x, 0) is replaced by - min ∂t u + F(x, 0, u, I u, Du, D2u), u − uo ≤ 0 in viscosity sense at (x, 0), - property (7) is satisfied at (x, 0),


5

at all x ∈ R N . A lower semicontinuous, locally bounded function u is a supersolution for (1), (2) with generalized initial condition, GIC, if the initial condition u(x, 0) ≥ uo (x, 0) is replaced by P 2− u(x, 0) = 0/ whenever u(x, 0) < uo (x, 0). A sub/supersolution with generalized initial condition is indeed a sub/supersolution in the sense of Definition 2. Lemma 1 If u is a subsolution (resp., a supersolution) for (1), (2) with GIC, then u(x, 0) ≤ uo (x, 0) (resp. u(x, 0) ≥ uo (x, 0)) for all x ∈ R N . Proof We first consider the subsolution case and suppose by contradiction that u(xo , 0) > uo (xo , 0) at some xo ∈ R N . By semicontinuity u > uo near at (xo , 0), hence ∂t u + F(x,t, u, I u, Du, D2u) ≤ 0 and the proof follows as in [3, Lemma 3.3]. With respect to the supersolution case, if by contradiction u(xo , 0) < uo (xo , 0) at some xo ∈ R N we look at the family of test functions n ϕn (x,t) = − x − xo 2 − nt. 2 By standard semicontinuity arguments, u − ϕn has a minimum point (xn ,tn ) on the compact set x : x − xo ≤ 1 × [0, T ] and we may suppose without loss of generality that (xn ,tn , u(xn ,tn )) → (xo , 0, u(xo , 0)). In particular P 2− u(xn ,tn ) is not empty, because it contains (∂t ϕn Dϕn , D2 ϕn ) computed at (xn ,tn ). Therefore u(xn ,tn) ≥ uo (xn ,tn ) even if tn = 0, by definition of supersolution with GIC. Eventually passing to the limit gives u(xo , 0) = lim u(xn ,tn ) ≥ lim uo (xn ,tn ) = n→∞

n→∞

uo (xo , 0), which is in contrast with the former assumption.

2.1 Penalization of the obstacle We are now ready to introduce the family of penalized equation 1 ∂t u + F(x,t, u, I u, Du, D2u) = (uo − u)+ , (x,t) ∈ R N ×(0, T ], (8) ε for any given ε > 0. Suppose that any of the standard Cauchy problems (8), (2) have a solution uε : then naive intuition suggests that u = lim uε satisfies u ≥ uo ε →0

and ∂t u + F(x,t, u, I u, Du, D2u) = 0 if u > uo , namely that it solves the obstacle problem. Unfortunately, it is not obvious to suppose, in general, that such a limit function u does exist, nor that, whenever it exists, it is continuous. A powerful tool to overcome this difficulty stands in the technique of half-relaxed limits introduced by [5], consisting in splitting the problem into two parts. In the first one upper and lower limits for the approximating uε are separately introduced: u (x,t) =

lim sup

uε (ξ , τ ), u (x,t) =

(ε ,ξ ,τ )→(0,x,t)

lim inf

uε (ξ , τ ).

(ε ,ξ ,τ )→(0,x,t)

(9)

Since u and u are respectively upper/lower semicontinuous and their parabolic semi-jets can be approximated with the ones of uε , they are expected to be respectively sub/supersolution. In our setting the stability of the integral terms I uε is

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A.L. Amadori

not trivial: it can be obtained by uniformly dominating M uε by means of some function of Bx,t . The second step consists into establishing a comparison principle for the final problem - the obstacle one - and in noticing that this implies the coincidence of the two limit functions. We end this section by performing the first step, namely by establishing that the penalization method produces a subsolution and a supersolution to the obstacle problem. Theorem 1 (Stability of penalization) We suppose that, for all ε > 0, there exists a viscosity solution uε for the Cauchy problem (8), (2) fulfilling (i) uε are locally bounded, uniformly with respect to ε , (ii) for all (x,t) ∈ R N ×[0, T ] there exist a neighborhood V and a function Φ ∈ C (R N ) ∩ L1 R N ; μx,t such that:

lim

(y,s)→(x,t)

Φ (z) d μy,s (z) =

Φ (z) d μx,t (z),

−Φ (z) ≤ M(h(y + z, s), w) ≤ M(k(y + z, s), w) ≤ Φ (z), for all (y, s) ∈ V , for almost all z ∈ R N (according to the measure μy,s ), and for all w in the closure of the set uε (y, s) : (y, s) ∈ V, ε > 0 . Then u and u , defined in (9), are respectively sub/supersolution for the obstacle problem (1), (2). Proof We begin by showing that u is a subsolution; by Lemma 1, it suffices to show that it is a subsolution with GIC. To this purpose, we fix (x,t) ∈ R N × [0, T ] with u (x,t) > uo (x,t) (otherwise, there is nothing to prove) and (τ , p, X) ∈ P 2+ u (x,t): we want to show that τ + F(x,t, u , I u , p, X) ≤ 0. One may easily find a sequence (εn , xn ,tn ) converging to (0, x,t) and (τn , pn , Xn ) ∈ P 2+ uεn (xn ,tn ) such that (uεn (xn ,tn ), τn , pn , Xn) tends to (u (x,t), τ , p, X). In particular we may suppose that uεn (xn ,tn ) > uo (xn ,tn ). Because uεn are subsolutions for (8), (2) , necessarily tn > 0 and

τn + F(xn ,tn , uεn , I uεn , pn , Xn ) ≤

+ 1 uo (xn ,tn ) − uεn (xn ,tn) = 0. εn

By passing to the limit, F.1 and F.2 give

τ + F x,t, u , lim sup I uεn (xn ,tn ), p, X ≤ 0. n→∞

Due to assumption F.2, the proof can be completed by showing that lim sup I uεn (xn ,tn ) ≤ I u (x,t). n→∞

It is an easy consequence of hypothesis (ii) and [1, Lemma 1] that lim sup I uεn (xn ,tn ) ≤ n→∞

R

N

lim sup M (uεn (xn +z,tn ), uεn (xn ,tn )) d μx,t (z). n→∞


7

Therefore, because lim sup uεn (xn + z,tn ) ≤ u (x + z,t) by definition, the monon→∞

tonicity of M yields the desired inequality. Next we prove that u is a supersolution with GIC. If u (x,t) ≥ uo (x,t) and t > 0 (otherwise, there is nothing to prove), we may conclude exactly as in the previous step. Otherwise, if u (x,t) < uo (x,t) and t ∈ [0, T ], we have to show that P 2− u (x,t) = 0. / Let us suppose by contradiction that there exists (τ , p, X) ∈ P 2− u (x,t). By standard arguments in the theory of viscosity solutions, x,t, u (x,t), τ , p, X can be approximated, respectively, by xn ,tn , uεn (xn ,tn ), τn , pn , Xn , where uεn solve (1.εn )-(2) and (τn , pn , Xn ) ∈ P 2− uεn (xn ,tn ). Moreover, since u (x,t) < uo (x,t), we may suppose without loss of generality that uεn (xn ,tn ) ≤ uo (xn ,tn ) − [uo (x,t) − u (x,t)]/2. Thus tn > 0 and 1 (uo (xn ,tn) − uεn (xn ,tn ))+ εn 1 ≥ (uo (x,t) − u (x,t)) . 2εn

τn + F(xn ,tn , uεn , I uεn , pn , Xn ) ≥

Eventually we obtain a contradiction by passing to the limit as n → ∞, because the first term is bounded by hypothesis (ii), while the last one blows up.

3 Comparison principle and existence of solutions It is well known that comparison principle among semicontinuous sub/supersolutions plays a crucial role in the viscosity solutions approach, because it allows to obtain existence (and uniqueness) of solutions, if combined with approximation arguments. Here, we first establish a strong comparison principle for the obstacle problem, next we show that the solutions to the penalized problems (8), (2) actually satisfy the hypotheses of Theorem 1. As a byproduct, we get well-posedness of the obstacle problem. By now we deal with the particular class of semilinear integro-differential equations given by (4), which includes many of financial interest. In view of the financial application, we take account of exponentially growing obstacles: H.0 there exists an integer no such that e−no x uo ∈ C R N × [0, T ] ∩ L∞ R N × (0, T ) . Moreover we assume that the linear integro-differential operator LI is of type 1 LI u = − tr σ σ T D2 u + b · Du + c u − d I u, 2 where σ = (σi j ) is a N × D matrix, b = (b1 , · · · , bN ), c, d are scalar numbers and H.1 σi j , b j , c are bounded and continuous functions of x,t, and σi j , b j are locally Lipschitz continuous w.r.t. x (uniformly w.r.t. t); moreover d ≥ 0 H.2 The function H is continuous with respect to all the variables and monotone non-increasing with respect to the fourth one. For all R > 0, there exists a modulus of continuity ωR such that H(x,t, u, I, p) − H(y,t, u,I, p) ≤ ωR ((1 + |u| + |I| + |p|)|x − y|) (10)

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A.L. Amadori

if |x|, |y| ≤ R. Moreover, there exists L ≥ 0 such that H(x,t, u, I, p) − H(x,t,v,J,q) ≤ L [|u − v| + |I − J| + |p − q|],

(11)

for all (x,t) ∈ R N×[0, T ], u, v, I, J ∈ R, and p, q ∈ R N . Lastly, there exists n1 ≥ 0 such that e−n1 |x| H(x,t, 0, 0, 0) ∈ C (R N × [0, T ]) ∩ L∞ (R N × (0, T )).

(12)

H.3 There exists n ≥ max(no , n1 ) such that the function

N(x,t) :=

1 + en|z| d μx,t (z)

is bounded and continuous. Adapting the technique used in [3, Theorem 3.4] for the Cauchy problem, the following comparison principle is attained. Theorem 2 (Comparison Principle) We consider the semilinear case (4) under assumptions H.0–H.3, and take u and u, respectively, a semicontinuous sub and supersolution to (1), (2) such that e−nx u, e−nx u ∈ L∞ (R N × (0, T )), where n is the integer appearing hypothesis H.3. Then u ≤ u in R N × [0, T ]. Proof Without loss of generality, we may replace the norm x with (x) = 1 + x2 , with the advantage that is of class C 2 . By hypothesis v(x,t) = e−n(x) u(x,t) and v(y,t) = e−n(y) u(y,t) are bounded sub/supersolution for the modified problem

(13) min ∂t v + LI v + H x,t, v, I v, Dv , u−e−n uo = 0, v(x, 0) = e−n(x) uo (x).

(14)

Here, we have used the notations 1 LI v := − tr σ σ T D2 v + b · Dv + cv − d I v, 2 where b := b − 12 n D σ σ T and c := c + n b · D − 12 n tr σ σ T D2 − 12 n2 Dσ 2 , H (x,t, v, I v, Dv) = e−n(x) H(x,t, en(x)v, I v, en(x)(nvD(x) + Dv)), and I v(x,t) :=

M en(x+z)v(x+z,t), en(x) v(x,t) μxt (dz).

Theorem 2 is equivalent to the comparison principle among bounded sub/supersolutions to (13)–(14). We set θ (t) = sup (v(x,t) − v(x,t))+ ‘ : x ∈ R N


9

and we prove the thesis by checking that θ ≡ 0. By making use of a remark by Ishii and Kobayasi [11, Theorem 3, page 918] (see also [3, Lemma 3.5]), it suffices to show that its upper semicontinuous envelope θ ∗ fulfills in viscosity sense min θ ∗ − C θ ∗ , θ ∗ ≤ 0, t ∈ [0, T ], (15) for a suitable constant C = (1+n)L − inf c + (L − inf d) DM∞ N∞ . To this aim we fix to ∈ [0, T ]: if θ ∗ (to ) = 0 there is nothing to prove. Otherwise, if θ ∗ (to ) > 0, we take a smooth function ϕ (t) touching from above the graph of θ ∗ at to . We want to take advantage from the fact that v and v are sub/supersolutions of the integro-differential inequality (13) to achieve an estimate from above of ϕ (to ), which does not depend on ϕ nor on to . To this end we proceed in the standard way by defining the auxiliary function, depending on two positive parameters α and δ ,

Φα δ (x, y,t) = v(x,t) − v(y,t) −

α δ x − y2 − x2 − ϕ (t). 2 2

It is straightforward to check that, for all fixed α , δ , there exists (xα δ , yα δ ,tα δ ) at which Φα δ has a maximum on the set R N × R N × [0, T ]. By extracting possibly a subsequence, (xα δ , yα δ ,tα δ ) converge to (xδ , xδ ,tδ ) as α → ∞, where (xδ ,tδ ) is a maximum point of the function v(x,t) − v(x,t) − δ2 x2 − ϕ (t) on the set R N × R N × [0, T ]. Moreover we have that

α δ 2 2 lim lim xα δ − yα δ + xα δ = 0, lim lim tα δ = to , 2 δ →0 α →+∞ 2 δ →0 α →+∞ lim lim v(xα δ ,tδ ) − v(yα δ ,tα δ ) = θ ∗ (to ). δ →0 α →+∞

Next, we apply [8, Theorem 8.3] with ε = 1/α to v1 (x,t) = v(x,t) − δ2 x2 , v2 (y,t) = v(y,t), φ (x, y,t) = α2 x − y2 + ϕ (t) . It yields that there are τ1 , τ2 ∈ R and two symmetrical matrices X, Y (depending on α , δ ) such that 2+ τ1 , α (xα δ − yα δ ) + δ xα δ , X + δ I ∈ P v(xα δ ,tα δ ), 2− τ2 , α (xα δ − yα δ ) , Y ∈ P v(yα δ ,tα δ ),

τ1 − τ2 = ϕ (tα δ ),

X 0 I −I ≤ 3α . 0 −Y −I I Since v and v are a sub/supersolution of (13), we get (omitting the dependence from α , δ ,t)

min ϕ + F x, v, I v, α (x−y)+δ x, X+δ I − F y, v, I v, α (x−y), Y , , v(x) − v(y) − e−n(x) uo (x) − e−n(y) uo (y) ≤ 0,

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A.L. Amadori

where we have used the abbreviated notation 1 F (x,t, v, I, p, X) = − tr σ σ T X + b · p + cv − d I + H (x,t, v, I, p). 2 Passing to the limit, the continuity of uo and the choice of the approximating sequence (xα δ , yα δ ,tα δ ) give

lim lim v(x,t) − v(y,t) − e−n(x) uo (x) − e−n(y) uo (y) = θ ∗ (to ) > 0. δ →0 α →+∞

Hence

lim sup lim sup ϕ + F (x, v, I v, α (x−y)+δ x, X+δ I) δ →0 α →+∞ −F (t(y, v, I v, α (x−y), Y) ≤ 0.

Eventually, the same arguments of [3, Theorem 3.4] show that ϕ (to ) ≤ C ϕ (to ); so (15) is obtained and the thesis follows.

We are now ready to state existence and uniqueness of solutions, as a corollary of Theorems 1 and 2. It remains to check that the solutions of the penalized problems (16) satisfy the assumptions of Theorem 1: it can be done by invoking the comparison result [3, Theorem 3.4]. Theorem 3 (Existence and uniqueness) Under assumptions H.0–H.3, the obstacle problem (1), (2) admits a continuous viscosity solution u fulfilling e−max(no ,n1 )x u(x,t) ∈ L∞ (R N × (0, T )), which is unique in the class

v ∈ C R N × [0, T ] : e−nx v(x,t) ∈ L∞ (R N × (0, T )) . Proof For any fixed ε > 0, the penalized equation 1 ∂t u + LI u + H(x,t, u, I u, Du, ) = (uo − u)+ , ε

(16)

fulfills the hypotheses of [3, Theorem 1.1]. Hence the Cauchy problem (16), (2) has an unique viscosity solution uε . Besides, it is an easy exercise to show that there are three positive constants λ , A , B such that h(x,t) = −eλ t At + B emax(no ,n1 )(x) and

k(x,t) = eλ t

At + B e

max(no ,n1 )(x)

are respectively classical sub/supersolution for all problems (16)–(2). Therefore the comparison principle [3, Theorem 3.2] guarantees that |uε (x,t)| ≤ eλ T (A T + B)emax(no ,n1 )(x) .


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Moreover, in view of assumption H.3, the function Φx,t (z) = eλ T (A T + B)emax(no ,n1 )(x) 1 + emax(no ,n1 )(z) satisfies assumption (ii) of Theorem 1 at all x,t. So, we may apply Theorem 1 and we obtain that u and u , defined in (9), are respectively a sub/supersolution. Next, Theorem 2 yields that u ≤ u on R N × [0, T ]. Therefore, since the inverse inequality holds by construction, u = u is a continuous viscosity solution, indeed. Uniqueness follows immediately from the comparison principle 2.

4 Further regularity In this section we investigate the regularity of the solution with respect to x. Because the operator (4) reduces to a first order one when σ = 0, the best regularity that may be attained is the same of the obstacle uo . On the other hand, in the financial application uo is only locally Lipschitz continuous with respect to x, namely H’.0 there exists an integer n such that

e−nuo ∈ C (R N × [0, T ]) ∩ L∞ 0, T ;W 1,∞ (R N ) .

The other hypotheses can be strengthen as follows. H’.1 Hypothesis H.1 holds and, moreover, σi j , b j , c, and d are globally Lipschitz continuous as functions of x, uniformly with respect to t. H’.2 Condition (11), (12) still hold, while (10) is strengthened by (17) H(x,t, u, I, p) − H(y,t, u,I, p) ≤ L 1+|u|+|I|+|p| |x − y|,

for all x, y, p ∈ R N , t ∈ [0, T ],u, I ∈ R. With respect to the integral term, we assume that the measures μx,t have the following structure d μx,t (z) =

M

∑ ν k (x,t) δ (ζ k (x,t) − z) + ν˜ (x,t; z) dz,

k=1

where δ and dz stand, respectively, for the Dirac and the Lebesgue measure. Net we assume H’.3 the functions ν k and ζ k are bounded, continuous, and Lipschitz continuous w.r.t. x, uniformly in t. Moreover there exists D > 0 such that 1 + en|z| |ν˜ (x,t; z) − ν˜ (y,t; z)|dz ≤ D |x − y|.

We shall take advantage of the fact that u has been constructed by the penalization method: since u is the pointwise limit of the solutions uε of the Cauchy problems (16), (2), uniform estimates of uε lead regularity on u. For instance, under assumption H’.0–H’.3 [3, Theorem 1.2] establishes that e−n uε ∈ L∞ 0, T ;W 1,∞(R N ) . Although the estimate of the Lipschitz constant which is provided blows up when ε goes to zero, so we need some refined argument to acquire the same result for the obstacle problem.

12

A.L. Amadori

Theorem 4 (Lipschitz continuity) Under assumptions H’.0–H’.3, the viscosity solution to (1), (2) is locally Lipschitz continuous with respect to x, precisely e−nx u ∈ L∞ 0, T ;W 1,∞ R N . Proof In order to prove that v = e−nu belongs to L∞ 0, T ;W 1,∞(R N ) , we set B = De−n uo ∞ , and vε = e−n uε , where uε is the viscosity solution of the penalized Cauchy problem (1ε ), (2). Since by Theorem 3 v(x,t) =

lim

(ε ,ξ ,τ )→(0,x,t)

vε (ξ , τ ),

the thesis may be achieved by showing that there exists λ ≥ 0 not depending on ε such that

ϑε (t) = sup vε (x,t) − vε (y,t) − Beλ t x − y : x, y ∈ R N is equal to 0 for all t and ε . By invoking [3, Lemma 3.2], it suffices to show that its upper semicontinuous envelope ϑε∗ fulfills min ϑε∗ − C ϑε∗ , ϑε∗ ≤ 0, for all t ∈ [0, T ], (18) for a suitable constant C. This may be done by making use of the fact that any vε solves in viscosity sense 1 + e−n uo − v , ∂t v + LI v + H x,t, v, I v, Dv = (19) ε and take the initial value (14). Here, we have used the same notations of (13). Let us fix to ∈ [0, T ]: if ϑε∗ (to ) = 0, assertion (18) plainly holds. Otherwise, if ϑε∗ (to ) > 0, we fix a smooth function ϕ such that (ϑε∗ − ϕ )(to ) = 0 is a strict maximum. It is straightforward to check that there exists a sequence (xδ , yδ ,tδ ) (depending on ε too) of points of local maximum for the function

δ Φδ (x, y,t) = vε (x,t) − vε (y,t) − Beλ t x − y − x2 − ϕ (t) 2 such that, up to an extracted subsequence, lim

δ →0

δ x 2 = 0, 2 δ

lim tδ = to ,

δ →0

lim vε (xδ ,tδ ) − vε (yδ ,tδ ) − Beλ tδ xδ − yδ = ϑε∗ (to ) > 0.

δ →0

(20) (21)

As a consequence, we may suppose that tδ > 0 (because of H’.0) and that xδ −yδ is bounded away from zero (because vε is continuous by Theorem 3). On the other hand, by applying [8, Theorem 8.3] for ε = B eλ tδ xδ − yδ > 0 to v1 (x,t) = vε (x,t) + δ2 x2 , v2 (y,t) = vε (y,t) and φ (x, y,t) = Beλ t x − y + ϕ (t) and making the same computations of [3, Theorem 1.2] we get

ϕ (tδ ) ≤ Cϑε∗ (tδ ) +

C

−λ

Be

λ tδ

1 xδ − yδ + o(1) + hε (xδ , yδ ,tδ ) , ε


13

as δ → 0, with + + hε (x, y,t) = e−n(x) uo (x,t) − vε (x,t) − e−n(y) uo (y,t) − vε (y,t) . Here, C and C only depend by the essential supremum over t ∈ [0, T ] of the W 1,∞ norm of σ , b, c, d, ν , ζ , and on the parameters appearing in H’.2 and H’.3, but not by ε . Eventually, if lim inf hε (xδ , yδ ,tδ ) ≤ 0 δ →0

(22)

for all fixed ε > 0, by choosing λ = C and passing to the limit as δ → 0 we get (18) and so the thesis. But (22) trivially holds if lim e−n(xδ ) uo (xδ ,tδ ) − vε (xδ ,tδ ))+ ≤ 0. Otherwise by H’.0 we have

δ →0

hε (xδ , yδ ,tδ ) ≤ − vε (xδ ,tδ ) − vε (yδ ,tδ ) − Beλ tδ xδ − yδ .

Hence (22) is implied by (21)

5 Application to the pricing of American claims In this section, we apply the obtained results to the pricing of American derivatives in jump-diffusion markets with large investor. We refer to [17] and to the references therein for a general presentation of jump-diffusion processes in market modeling. We also cite [15], [16], and [13], about the relation among obstacle problems in the viscosity setting and American claims in jump-diffusion markets. In what follows we take into account N risky assets whose prices vector T S = S1 , · · · , SN evolves according to the stochastic differential equation dSt = St − α (St ,t)dt + σ (St ,t)dWt + γ (St ,t, z)ν (dz, dt) , where Wt stands for a D -dimensional Brownian, ν (dz, dt) stands for a Poisson random measure, S = diag (S1 , · · · , SN ), α = (α1 , · · · , αN)T is the drift vector, σ = (σi j ) i = 1,· · ·, is the diffusion matrix, and N

j = 1,· · ·, D

γ = (γ1 , · · · , γN )T is the jump vector.

(23)

14

A.L. Amadori

In the usual Black & Scholes model, the price U of an American claim is given by a deterministic function of the free variables S and t, which solves an obstacle problem plus a final condition: min{−∂t U + F [U ],U − G} = 0 U (S, T ) = G(S)

N S ∈ R+ , t ∈ [0, T ), N S ∈ R+ .

(24) (25)

The function G and the final time T are input data, standing respectively for the payoff and the expiration time of the claim. The operator F is, in this case, linear and elliptic. The paper [15] shows that this holds true also in the jump-diffusion setting, provided that we take into account an integro-differential operator 1 F [U ] = − tr (Sσ )(Sσ )T D2U + r (U − S · DU ) − J [U ], 2 where the integro-differential term J [U ] is the risk-neutral contribution of the jumps to the price: J [U ](S,t) = λ

[U (S + Sγ ,t) −U (S,t) − Sγ · DU (S,t)] φ m(dz).

(26)

The quantity φ is known as the “market price for (jump) risk”, and it is a positive function of S, t, and z. In general it is not uniquely determined, because of the incompleteness of the market. By the way, in that particular cases where the market is complete, and the market price for risk stays uniquely determined, we have that rank σ < N, so that the second order term is strongly degenerate, see [12]. The paper [9] shows that the presence of a large investor, acting in the underlying market, causes non-linear effects into the elliptic operator F . For instance, in [3] the interest rate r is allowed to vary according to the amount invested into the bank. After computations, one gets the following non-linear integro-differential operator 1 F [U ] = − tr (Sσ )(Sσ )T D2U − J1 [U ] 2 + r(S,t,U − S · DU − J2 [U ]) (U − S · DU − J2 [U ]),

(27)

where J1 and J2 are integro-differential operators of type (26) which only differ for the weight function φ . With this choice of F , the obstacle problem (24), (25) is not included in the class studied by [16], but it can be led to the class studied in this paper. In view of applying our result to this problem, we state precise hypotheses. Standard assumptions about the market coefficients are A.1 For any j = 1, · · · , N and i = 1, · · · , D, the functions α j and σi j are continuous w.r.t S and t with the regularity N N N f ∈ L∞ 0, T ;W 1,∞ (R+ ) , S D f ∈ L∞ (R+ × [0, T ]) ,

for f equal to α j , σi j .


15

A.2 The intensity of the Poisson random measure ν (dz, dt) decomposes in λ (t) m(dz) dt, where λ > 0 is a continuous function of t, and m is a probability measure. As j = 1, · · · , N and k = 1, 2, the functions γ j and φk are continuous w.r.t S and t, and Borel-measurable w.r.t. z; the regularity property N N N f ∈ L∞ 0, T ;W 1,∞ (R+ ) , S D f ∈ L∞ (R+ × [0, T ]) ,

holds for f = γ j (·, ·, z) (with j = 1, · · · , N), φk (·, ·, z), uniformly with respect to z. Moreover φk > 0 and there is ρ > 0 so that 1 + γ j ≥ ρ > 0 Concerning the interest rate and the large investor assumption, we take N A.3 r is a non-negative continuous function of S,t, ξ on R+ × [0, T ] × R\{0}. For all ξ ∈ R \ {0}, the function (S,t) → r(S,t, ξ ) satisfies N N N r(·, ·, ξ ) ∈ L∞ 0, T ;W 1,∞ (R+ ) , S Dr(·, ·, ξ ) ∈ L∞ (R+ × [0, T ]) , N uniformly with respect to ξ . Moreover for all (S,t) ∈ R+ × [0, T ], the function ξ → r(S,t, ξ )ξ is non-decreasing and Lipschitz continuous on R, uniformly with respect to (S,t).

We refer to the same paper [9] for some concrete examples fulfilling assumption A.3. Next we explicitly mention that the typical pay-off function G has, at most, polynomial (often linear) growth for large values of S. The main result of this section can be stated as follows. Theorem 5 We assume that the market parameters satisfy assumptions A.1–A.3, and consider the obstacle problem (24), (25) with F given by (27). This problem is well posed in the class of continuous functions with polynomial growth for large N S. In particular, for all payoffs G ∈ C (R+ ) with 0 ≤ G(S) ≤ B o (1 + Sno ), the solution satisfies

0 ≤ U (S,t) ≤ B (1 + Sno ). Moreover, let Uε the solution to the problem with penalization: 1 (G −Uε )+ ε Uε (S, T ) = G(S)

−∂t Uε + F [Uε ] =

N S ∈ R+ , t ∈ [0, T ),

(28)

N S ∈ R+ .

(29)

Then, as ε → 0, Uε tends to U pointwise and in the following sense: U (S,t) =

lim

(S ,t )→(S,t) ε →0

Uε (S ,t ).

If, in addition, the payoff G is locally Lipschitz continuous w.r.t.S with |G(S)| + DG(S) ≤ B o (1 + Sn + S−n ), the solution of (24), (25) is locally Lipschitz continuous with respect to S, namely |U (S,t)| + DU (S,t) ≤ B (1 + Sn + S−n ) N for all (S,t) ∈ R+ × (0, T ) .

16

A.L. Amadori

Proof We perform the logarithmic change of variable xi = log Si and introduce the function u(x,t) = U (S(x), T −t). The problem (24), (25) for U (S,t) is mapped into an obstacle problem of type (1), (2) for u(x,t), of semilinear type (4). Actually, denoting f˜(x,t) = f (S(x), T − t) for f = σi j , γ j (·, ·, z), λ , φk (·, ·, z), r(·, ·, ξ ), we have 1 F(x,t, u, I u, Du, D2u) = − tr σ˜ σ˜ T D2 u + b˜ · Du − λ˜ I1 u + H(x,t, u, I2u, Du) 2 H(x,t, u, I2u, Du) =˜r (x,t, u + c˜ · Du − λ˜ I2 u) (u + c˜ · Du − λ˜ I2 u) with 1 b˜ j = ∑ σ˜ 2jk + 2 k c˜ j = Ik u(x,t) =

γ˜ j φ˜1 m(dz)

γ˜ j φ˜2 m(dz) − 1 k [u(x + z,t) − u(x,t)] μx,t (dz),

j = 1, · · · , N, j = 1, · · · , N, k = 1, 2.

k Here the measures μx,t (dz) are assigned by means of

k ϕ (z)μx,t (dz) =

ϕ log(1 + γ˜1 ), · · · , log(1 + γÑ ) φ˜k m(dz),

for all continuous, bounded function ϕ . It is clear that hypotheses A.1–3 imply conditions H.1–3 (with n = 0). Therefore the existence and uniqueness of the solution, its pointwise estimate and its approximation by penalization hold true by Theorem 3. Concerning the Lipschitz-regularity and the first order estimate, it follows by Theorem 4. Indeed, assumption H’.3 does not hold for general measures m, but one can check that the result is still effective by running through the computations of [3, Theorem 1.2].

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